Jan 7, 1986 - Mori theory [10] and observations of suitable P^fibrations and elliptic ... (i) V* is either P2 or a ruled surface Fm(m>0) with a Pl-fibration v: Fm.
Zhang, D. Osaka J. Math. 24 (1987), 417-460
ON IITAKA SURFACES DE-QI
ZHANG
(Received January 7, 1986) Introduction. Let k be an algebraically closed field of characteristic zero. We consider a pair (V, D) which satisfies the following conditions: ( i ) V is a nonsingular, projective and rational surface defined over k and D is a reduced effective divisor on V with simple normal crossings; (ii) (V, D) is almost minimal; (iii) K(V-D)=0 andpg(V-D):=dim H°(V, D+Kv)=l. We shall call such a pair (F, D) an Iitaka surface. A surface of this kind has been studied by Iitaka [4]. Thence comes the naming of Iitaka surface. In Th. 3 [ibid.], he gave an explicit way of writing down possible configurations of the divisor D. However, he did not determine which of these configurations are realizable. To begin with, he did not employ our almost minimal model to classify such surfaces. Since an almost minimal model in the context of non-complete surfaces is thought as a substitute of a minimal surface in the context of complete surfaces, it would be natural to include the almost minimality in the definition of Iitaka surfaces. Thanks to this definition, we can determine (and classify) all Iitaka surfaces. Our method depends heavily on the theory of peeling in [9], the Mori theory [10] and observations of suitable P^fibrations and elliptic fibrations. Our Main Theorem consists of the following two results: Reduction Theorem. Let (V,D) be an Iitaka surface. Then the following assertions hold true. (1) There exists a unique decomposition D=A-\-N with A>0 and i\T^0 such that A-\-Kv>—'0, N is disjoint from A and the connected components of N consist of (—2) rods and (—2) forks (cf. the terminology below). (2) There exists a birational morphism from V to a minimal rational surface V*> u:V-+V* satisfying the following conditions: ( i ) V* is either P2 or a ruled surface Fm(m>0) with a Pl-fibration v: Fm 1 -+P . Moreover, A*:=u*A is a divisor with (at worst) normal crossing singularities and A*-\-Kv*~0.
418
D.-Q. ZHANG
(ii) Suppose V*=P2. Then u*D=u*A. (iii) Suppose V*=Fm. Let M be a minimal section of F * and let /,• (1 ^i^ n;n^4-) be all fiber of v such ihatf{ D A* consists of one smooth point of A*. There exist a fiber hx ofv, a nonsingular rational curve Cx with (C\)=2 or 4 and a nodal rational curve C2 with C 2 G | — Kv* \3 such that Ai=t=/t- (1 ^i^n), hu Cx and C2 are not components of A* and that D*:=u*D is apart of A*+fx-\ Vfn+M+^+Cx + C2. The curves hu Cx and C2 are specified in the next condition. (iv) If hlf Cx or C2 appears in D*, then A* is either an elliptic curve or a nodal curve and Z)* has one of the following nine configurations; where m^l in Fig. 7 and Fig. 8 below and m=2 otherwise and, A* is an elliptic curve in Fig. 6, Fig. 7 and Fig. 8. ( v ) If M is a component of D*y then m>2. (vi) If ni^Zy then D* is given in Lemma 2.6.
K Fig. 1
-2 M
2 0U
0h Fig. 3
Fig. 4
Fig. 5
Fig. 6
-2
M 2
419
IITAKA SURFACES
f2 Fig. 8
Fig. 7
-2 M
®^ X
-§• A*
2
Fig. 9
Existence Theorem. (1) Let (V,D) be an Iitaka surface with Consider the following operations on D: 1i) Let P be a smooth point of A and letw: V-+V be a sequence of blowingups with center at P and its n(n^0) infinitely near points lying consecutively on the proper transforms of A. Let R:=W~\P)—(the last (—1) curve) which is a (—2) rod with n components. Let A':=w'A be the proper transform of A, let N':=w*N+R and let D':=A'+N'. (ii) Let P be a double point of A and let w\V'->V be the blowing-up with center at P. Let A'l^w^A, N':=w*N and D':=A'+N'. (iii) Suppose that there exists a (—1) curve E on V such that any connected component of E+N has either a rod or a fork as its dual graph. Let P:=A f] E and let w: V'->V be the blowing-up of P. Let A'\=w'Ay N':=w'E+w*N and
D':=A'+N'. Let (V, D') be a pair obtained from (V> D) by performing finitely many operations of type (i), (ii) or (iii) on D. Then (V',Df) is an Iitaka surface. (2) Let (F*, D*) be a pair as in Reduction Theorem. A minimal resolution of (V*,D*) is, by definition, the shortest sequence of blowing-ups u: VQ->V* such that u~lD* is a divisor with simple normal crossings. Let Do be a reduced effective divisor obtained from u~lD* removing all (—1) curves except for the (—1) curve arising from a possible, unique node of A*. Then the pair (Vo, Do) is an Iitaka surface. (3) Every Iitaka surface (V,D) is obtained from an Iitaka surface (V0,D0) as considered in the assertion (2) above by repeating the operations considered in
420
D.-Q. ZHANG
the assertion (1) above.
This paper consists of five sections. In §1, we shall consider under which conditions an litaka surface becomes a logarithmic K3-surface. At the begining of §2, we apply the theory of peeling and the Mori theory. By the first theory, we pass from an litaka surface (V, D) to a pair (P, D) by contracting BkD, where V is a projective normal surface with rational double points. We apply the Mori theory and show that we have only to consider three cases separately. Then we consider an litaka surface (V, D) with p ( F ) ^ 2 ; this will cover the first two cases. We treat the third case p(V)=l in §§3 and 4. Finally in §5 we consider complementary cases to complete the proof of Main Theorem. TERMINOLOGY. For the definitions of ^(logD) and the logarithmic Kodaira dimension K(V— D), we refer to litaka [3; Chap. 10 & Chap. 11]. For the definition of an almost minimal surface, we refer to [9; Sect. 1. 11], as well as the relevant definitions like the bark of Z), rods, twigs, forks, admissible twigs, rational rods, etc. By a (—i) curve we shall mean a nonsingular rational curve C with (C2) =—i (*'^1). By a (—2) rod (or (—2) fork, resp.) we shall mean a rod (or fork resp.) whose irreducible components are all (—2) curves. In other words, (—2) rods and (—2) forks have the weighted dual graphs of the minimal resolution of rational double points. A reduced effective divisor with simple normal crossings is abbreviated as an SNC divisor. K(V): the Kodaira dimension of V. the logarithmic Kodaira dimension of a nonsingular algebraic surface X defined over k. Kv: the canonical divisor of V. pg(V-D):=dim H°(V, D+Kv). q(V-D):=dim H°(V, O^{logD)). p(V): the Picard number of V. Fm: A minimally ruled rational surface on which there is a minimal section M with (M2)=—m. NOTATIONS.
K(X):
In the pictures of the configurations of curves (not the dual graphs), considered in our paper, if an encircled number appears, it means that two curves, between which the number is written, meet each other at a single point with the order of contact indicated by the number. I would like to thank Professor M. Miyanishi who gave me valuable suggestion during the preparation of the present paper. 1. Logarithmic K3-surfaces We shall begin with DEFINITION
1.1. Let (F, D) be a pair of a nonsingular projective surface
IITAKA SURFACES
421
V defined over k and a reduced effective divisor D with SNC (simple normal crossings) on V. We call this pair a log KZ-surface if the following conditions are met: (i) *(V-D)=0; (ii) the log geometric genus pg(V—D)=l; (iii) the log irregularity q(V-D):=dim H°(Vy nv(logD))=0. We hope to classify log i£3-surfaces (F, D) by looking into their almost minimal models (F, D). But (F, D) may not remain being a log i£3-surface. Indeed, (F, D) is an Iitaka surface (cf. [9; Lemma 1.10]), while the condition (iii) above may become false for (F, D). However, we have the following Lemma 1.2. Let {V,D) be a pair of a nonsingular protective surface V and an SNC divisor D on V. Let (F, D) be an almost minimal model of (V,D). Then we have: (1) / / K(V)=0 and (V,D) is a log KZ-surface, (F, J5) is also a log K3surface. (2) Conversely, if {VyD) is a log KZ-surface, then (V,D) is a log KZ-surface and either K(V) = — °O or K(V)=0. Proof. (1) Assume /c(V)=0. Then there exists an integer N>0 such that INKVI =hV be the birational morphism attached to an almost minimal model (F, D), where D=f*D. We know that q(V-D)=0 iff q(9)=0 and irreducible components of D are numerically independent (cf. Iitaka [4; Lemma 2]). We also know that pg(V-D)=pg(V-D) and £(F— D)=K(V— D) (cf. [9; Lemma 1.10]). Now assume that (F, D) is a log jO-surface. So, in order to verify the assertion (1) we have only to show that irreducible components of D are numerically independent. By inducting on the number of blowing-ups we have to perform to get (VyD) from (F, JS), we may assume that/is the contraction of a (—1) curve E on V (which means an exceptional curve of the first kind) such that: (a) (D*+Kv, E): = S?=iA ar*d Di:=f*Di for i=l, —, n. Then / * ( S 7 - i a / A ) = S ? - i « / A + « ' E = 0 for some a^iZ. We may assume a^O. If a=0, we get S?=i^,A = 0- So we have ^ = • • • ^ = 0 for q{V— D)=0 implies that Dly ••-,/)„ are numerically independent. Suppose that a>0. After a suitable permutation of {1, ••-,#}, we may assume that SJ-ial-A=Sf-i«,-A—T>ttj=s+ibjDj with a^O and &y^0. Then we get i0iA —2y-*+ify-Dy. Since q (V)=0, there exists an integer Nx>0
422
D.-Q. ZHANG
such that N^aE+^'i.iaiD^N^Sj.s+ibjDj. Let N2=Max{bs+u •••, bn}. By the assumption that a>0, we have iV2>0. Then N1N2D=N1N2(Dl-\
+N,N2(Dt+l+ - +D,)~N1N2{Dl+ .- +D (N2—bj)Dj. Since 1? appears in the right-hand side and does not appear in the left-hand side, we obtain dim\NlN2D\ > 0 . Since | JVK71 =t= ^, we have &\m\NlN2N(D+Kv)\^&im\NlN2ND\>Q, which is a contradiction because jc(V-D)=0. (2) It is easy. Q.E.D. The following result due to Kawamata [5] is crucial. Lemma 1.3. Let (V,D) be a pair of a nonsingular projective surface V and an SNC divisor D on V. Suppose that K(V—D)=0 and that (V,D) is almost minimal. Then n(D*-\-Kv)~0 for some Proof. See [6; Chap. II, Th. 2.2]. By using Lemma 1.3 and the results in [9], we verify the following lemma. Lemma 1.4. Suppose that (V,D) is a pair of a nonsingular projective surface V and an SNC divisor D on V. Suppose furthermore that K(V)=7C(V—D) =0 and that (V, D) is almost minimal. Thne the following are equivalent: (1) (V, D) is a log KZ-surface; (2) V is a minimal KZ-surface and D consists of (—2) rods and (—2) forks, where a (—2) rod (or (—2) fork, resp.) is a rod (or fork, resp.) whose irreducible components are (—2) curves, i.e., nonsingular rational curves with self-intersection (-2). (3) q(V)=0, pg(V)=l and D consists of ( - 2 ) rods and ( - 2 ) forks. Proof. Suppose that (F, D) is almost minimal and that K(V)=K(V— D) —0. Then, applying Lemma 1.3, we obtain nKv~0 for some n>0 and Z)*=0 since K(V)=0. So we know that SuppZ)=SuppBAZ), that D consists of (—2) rods and (—2) forks and that irreducible components of D are numerically independent. Hence q(V-D)=0 iff q(V)=0. We know that h°(V,n(D+Kv))= h\V,[nD^nKy\)=h\V,nKy) for every n>0 (cf. [9; Lemma 1.10]). Then Lemma 1.4 is obvious. Q.E.D. In the subsequent paragraphs of this section, we always assume that a pair (F, D) is an Iitaka surface. Then, since h°(V, [D*+Kv])=pg(V— D)=1, just one of the following two cases takes place. (1) There exists a curve A^[D*] w i t h ^ ( , 4 ) ^ 1 . (2) Every curve C^[D*] is rational and the dual graph of [D*] contains a rational loop A. Moreover, we can show
423
IITAKA SURFACES
Lemma 1.5. Let (V,D) be an Iitaka surface. Then A is a connected component of D with A-\-Kvr+~0 and IP=[IP]=A. Hence every connected component of D other than A is a (—2) rod or a (—2) fork. Furthermore, in case (1), we have pa(A)=l, i.e., A is an elliptic curve. Proof. Since \A+Kv\4=(j) both in the cases (1) and (2), we get Z)*= [D*]=A and A-\-Kv~0 by virtue of Lemma 1.3. Hence A is a connected component of D because Lfi=A implies that D contains no rational admissible twigs sprouting from A. In the case (1), we have pa(A)=l-\
(A+Kv,
A)=l. Q.E.D.
We know that the almost minimal model of a log K3 -surface is an Iitaka surface; see the remark before Lemma 1.2. Conversely, we have the following lemma. Lemma 1.6. Let (V, D) be an Iitaka surface. Then we have: (1) (V,D) is a log K3-surface provided that A is an elliptic curve. (2) / / A is a rational loop, there exists a birational morphism of pairs f: (V*,D*)->(V,D) such that (V*,D*) is a log KZ-surface, (V,D) is an almost minimal model of (V*,D*) and f is the associated morphism. Proof. (1) is obvious (cf. Lemma 1.5). (2) Suppose that (Vy D) is an Iitaka surface and that A is a rational loop. Let ux: VX->V be the blowing-ups of points Px and P2 on A as shown in the picture below. Let u2: V*->V1 be the blowing-ups of points Qx and Q2 on ufA
—u[A. Letf^u^Uz and D*=f(D)+C1+C2. that
Since A+Kv~0, it is easy to see We obtain easily D**=f'(A)+— 1 (C}+C2). Since -1
424
D.-Q. ZHANG
(D**+Kv*, E1)=(El+Ei+— (Ci+C2), E1)=-—* are numerically independent. Hence q(V*—D*)=0 for q(V*)=q(V)=Q. we know that ^ ( F * - D * ) = ^ ( F - D ) = 1 and ic(V*-D*)=tc{V-D)=O (cf. [9; Lemma 1.10]). So (F*, Z>*) is a log iO-surface. Therefore, the assertion (2) is verified. Q.E.D, We end this section with the following two lemmas. L e m m a 1.7. Let (V, D) be an Iitaka surface. If there exists a (—1) curve E on V, vie let u}: V->VX be the contraction cf E and let Ay=ux*A. Then Ax-\Kv r>^0 and Ax is an NC {normal crossings) divisor. Moreover, Ax is not an SNC divisor iff A is a loop consisting of two irreducible components, one of which is E.
Proof. Note that (A, £)=— (KVy E)=l, for A+Kv~0. obvious.
Lemma 1.7 is Q.E.D.
L e m m a 1.8. Suppose that (V,D) is an Iitaka surface. Then every nonsingular rational curve C on V has self-inter section more than (—3), unless C is a component of A.
Proof. Since ^ + £ ^ 0 , 0^(A, C) = (-Kv, C)=2-2pa(C)+(C2) = (C ), i.e., (C 2 )^— 2 for any nonsingular rational curve C with C ^ S u p p A Q.E.D. 2
2. Iitaka surfaces with f>(F)^2 Fix an Iitaka surface (F, D) in the present section. Let p: V-+V be the contraction of BkD. Then F is a projective normal surface with only rational double points as singularities and there exists an N^N such that NF is a Carrier divisor for every FeDiv(F) (cf. [9; Lemma 2.4]). Hence we have an intersection theory on V. Furthermore, we have Kv=p*Ky (cf. Artin [1; Th. 2.7]). We shall classify all Iitaka surfaces with DEFINITION 2.1. Let N(V): — {1 -cycles} Rf {numerical equivalence}, and let iVZ?(F):=the closure of the cone of effective 1-cycles {S?-i^i[CJ; Q : curve on F, [Q] where rKFmr^0 n\=§{singular fibers of O } . Case. A is an elliptic curve. Then fi(i=l, •••, n) passes through a ramification point of 7i I U^A. Hence n^A and k:=#{connected components of BkD} rgi2n^8. Case. A is a rational loop. Then m^l, A consists of a nonsingular fiber lx and a 2-section F, andfi(i=l> •••, n) passes through a ramification point of n \ U^F. Hence Proof. First of all, we shall construct a morphism : V-+P1 as in [9; Lemma 2.8]. Define rational numbers aly •••, ar by the condition: ( / + 2 ! - i a , A , Dj) = 0 for j = 1, ..., r where Supp BkD= U J-iZ),-* Since /^Supp BkD, we have a^O. We know that Nl is a Carrier divisor; see the definition of N before Definition 2.1. Evidently iaiDi) is supported by SuppBkD. So we have:
I^A))
by the definition of a/s and by (p*7,A)=0. because BkD is negative-definite.
Hence
= 0 p*Nl=N(l+^rimiaiDi)
IITAKA SURFACES
We know that h\V,np*Nl) = h°(Vy Kv-np*Nl) Riemann-Roch theorem we obtain:
427
= 0 for n>0.
So, by
h\V, np*Nl)^-j-(p*Nly KV)+X(OV) = - | ( M , Kv)+l->+oo as n->-\-oo because (/, Kv)PX such that &=&xoux. It is easy to see that ul*A-\-KVx B2)=0. In the case (i), we contract the unique (—1) curve u1*B1 in ul*fl and have one of the above two cases. Continue this process untill the case (ii) takes place. So, / 1 =2(2?+B 1 +"- +jB5_2)+jB$-i+i?s after a suitable change of indices {1, •••,$}. Its configuration is given in the statement of this lemma, where Di:=Bi. After the contraction of E,BX, •••,Z?S_1, the proper transforms of A and B9 meet each other in a single point with contact of order 2. We have seen that every connected component of BkD is contained in a singular fiber of . Hence, by the claim 4, we easily conclude Claim 5. Every connected component of BkD is a rod of type Au a rod of type A3 or a fork of type D8 (s^4) As in the proof of the claim 4, we contract all exceptional curves of the first kind contained in singular fibers of . Then we have a birational morphism u:V-+Fm onto a minimally ruled rational surface n\Fm->Pl such that Suppose BkD= of A*:=u*A and meets none of the other components of A*. Namely, the point t/*/i fl Af is a ramification point of 7t\A\'. Af-^P1, and Af is a 2-section of 7t\ Fm-^>Pl. This implies that the irreducible component Af of A* is uniquely determined. We know also that u does not contract any irreducible component of A (cf. the claim 3). Hence we have: # {irreducible components of A} = # {irreducible components of A*} We see easily that A* is an SNC divisor with A*+KFm~0 (cf. Lemma 1.7). We consider the following two cases separately to verify the remaining assertions of Lemma 2.5. Case. A is an elliptic curve. Let M* be a minimal cross-section of n: Fm->P1 and let /* be a general fiber of n. Then we have 0^(M*,i4*)=(Jlf*,-is: F J=(iM*, 2M*+(m+2)Z*)=2-m.
429
IITAKA SURFACES
Hence m^2. On the other hand, since (A*, /*)=2, TV\A*: A*->P1 is a double covering and hence it has exactly 4 ramification points. Thus, there are at most 8 connected components in BkD. Case. A is a rational loop. By the assumption that V^=Fmf we know that has a singular fiber. So, there is an irreducible component Af of A* such that 7t\A\: Af-^-P1 has a ramification point. On the other hand, we have mfLl as in the previous case. The case m=2 is excluded by virtue of Lemma 2.6 below. Thus, m=0 or 1, and A* consists of a 2-section Af and a fiber of by the same lemma. Now, counting the number of ramification points of a double covering zr | A*: Af-^P1, one knows that there are at most two singular fibers in and hence at most 4 connected components in BkD. Q.E.D. Lemma 2.6. Let (V,D) be an Iitaka surface. Suppose that V is isomorphic to P2 or Fm. Then the configuration of D is given as follows, where if V=Fm zee denote by M the minimal section and by I a general fiber. (1) Case. V=F2. Case (a) BkD^^. Then SuppBkD=M and the configuration of A is one of the following:
>M+2l
-2M+41
M+2/-X2
elliptic curve Case (b) BkD=. Then D=A of the following:
r
-2M+41
and the configuration of A is one
M+21
0
M
0
M+21
J
M+21
elliptic curve
-2
Z_
0
M+4/ 2
(2) Case. V=P .
Then BkD=^
and D=A.
The configuration of A is
430
D.-Q.
ZHANG
one of the following:
9~3H
2H~[4
elliptic curve where H is a line on P2. (3) Case. V=FQ. one of the following:
Then BkD= and D=A.
The configuration of A is 0
r 8 •2M+4Z
M+l
elliptic curve
o
/
0
M+21
A (4) Case. V=FV one of the following:
Then BkD=cj> and D=A.
The configuration of A is
1
8|~2M+3Z 2M+21 elliptic curve (5) Case. V=FM(tn^3). of A is one of the following:
Then BkD=(j> and D=A.
—m
i (M+ml) Proof.
Easy.
M+(m+2)l
The configuration
IITAKA SURFACES
431
3. Iitaka surfaces with />(F)=1, the part (I) In this section, we always assume that (V, D) is an Iitaka surface with p(V)=l. This case corresponds to the case where H=0. We begin with Lemma 3.1. Let (V,D) be an Iitaka surface with p(V)=l. Then vie have: ( i ) [A], [D{\, •••, [Dr] form a basis of N(V), where SuppBkD= U J-iA(ii) A is nefand,for any irreducible curve C on V, (A, C)=0 iff CciSupp BkD. In particular, every (—2) curve is contained in SuppBkD. (iii) (A2)^l. Hence r=9-(A2)^S. Furthermore, (A2)^6 if A is a rational loop. Proof. ( i ) is clear because p ( P ) = l . The assertion (ii) and the first part of the assertion (iii) are easy to verify. Note that A+Kv~0, p(V)+(Kv)=lO and p(V)=r+l. Hence we obtain r+(A2)=9. Suppose that A=Ax-\ \-At is a rational loop, where At is irreducible. We know that A is an SNC divisor, whence t^2. Since Af] SuppBkD= and since every irreducible curve on V is ample, we know that (A2-)^! for every i. Hence (^ 2 )=SLi(^?)+22,. Let ux: V->VX be the contraction of E. Write Supp.B&D:= U UiD{. By virtue of Lemma 3.1, we have p(V)=r+l, whence p(V^)=r. So, there exists (a, blf •••, br)^Rr+1—(0, •••, 0) such that
aul*A+*2jmibiul*Di=0. Since Ef]SuppBkD=^y we have u^Aftu? SuppBkD = (j> and hence 0=(%u4, au^A+^ri^ibiU^Dt)=a{u^A2)-=a{{A2)-\-\). Then we obtain a=0 because (^4 2 )+1^2 by virtue of Lemma 3.1. Since SJ=i%*A- i s obviously negative-definite, we must have b1=-"=br=O, which is a contradiction. Hence E meets BkD. Suppose that (E, AH [-Dr)=(E, R)=(E, D^l, where Dx is a tip of a rod R which is a connected component of BkD. We may write R=D1-{-"-+ Ds. Let W be the contraction of E+R. Since 0, because of the assumption p ( F ) = l and the condition (*). Let l~g'(l). Then / is a (—1) curve which either does not meet Bk(D—D^) or meets Bk(D—Z)j) in a single point on a tip Z)t- of a connected component of Bk{D—D^), which is a rod (cf. Remark 2.4). We consider these two cases separately.
Case (i-A) / meets Bk(D—D?). Case(i-K-?) / n A = f By virtue of Remark 2.4 and Lemma 3.2, / must meet F—Dly and F— D} is a rod, i.e., F is a fork of type D4. We see easily that Di=Dz or Z>4, say Di=D3. Then ^ ( Z + D a + i ^ + D H - D J gives us a P^fibration from V to P 1 . Note that BkD is contained in the fibers of 2)=1. The last condition is equivalent to x=n=l. Indeed, if Ln=0 and (/, A ) = l hold, then (Ln, l)=(Lny A ) = 0 . So we obtain — #— l+2rc=#—2n+rc = 0 , i.e., #=/z=l. Conversely, if x=n=l then .#2—(4ft—l)#+2ft2=0, whence Ln = 0 and (/, A ) —1- We show that x=l always. In fact, if we set n=2y we must have x2—7#+8>0, whence x^2. If we set n=4, we must have ) 2 —(24+—)>0.
(x—8-\ Li
Hence the only possible value for x is 1 because
T
and #4=2. So, we see Lx = 0. Since V is rational, we have Lx^ 0, i.e., A~l-\-Dx-\ hA- Thus the claim is verified. However, this is impossible because A~l-\-Dx-\ h A implies that AH + A is a connected component of BkD containing Dly while the connected component F(^DX) of BkD is a fork. Now we consider the next: Case. A is not a component of F. Then (I, Dx) = l, cf. the case (i-B-b) below. We shall show that this contradicts the condition (*) where we take A as A in the condition (*). Let h: V-+V2 be the contraction of Bk(D—Di). We obtain p(V2)=2 since p ( P ) = l . Let *=I2(/+D1+...+D,_2)+P,_1+D,_I_: F-^-P1. There exists clearly a morphism i/r: P2—>P* such that =i/ro/i. Let H=2h*ly which is a fiber of i/r. Then i\T(P2)=J2[A* A ] + « [ # ] • s i n c e (h*I, KV2)=-(h*l, h*A)= — 1, A*/ is an extremal rational curve. We easily see that H^ fl NE(V2)=R+[h*l]y H^O and (j3"2)=0. This contradicts the condition (*). Case (i-B) / does not meet Supp Bk(D—A). Then (/, A ) ^ l by virtue of Lemma 3.2. CVw*(i-B-a) ( / , A ) = 1 . This leads to a contradiction as in the case (i-A-a). Case (i-B-b) (/, A ) ^ 2 . Let Ln=(A2)l—A+nDx as in the case (i-A-b). Then we see that (A2)=ly (/? Dx)=2 and ^4-—l-\-Dx. Hence Dx is an isolated component of BkD, which is a contradiction. Therefore, we have proven that the case ( i ) does not occur. Case (ii) F is a fork of type E6. Let A be a component of F=Dx+D2-\ h A a s shown below: -2/
-2 -2
A
^ 4
2D.3
-2/
'A A
436
D.-Q.
ZHANG
As in the case(i), we apply the Mori theory to the surface Pj obtained from V by contracting Bk(D—D^), which is a rod. Case (ii-A) / meets Bk{D—D^). Case (ii-A-a) lf]D1=^. Then, by virtue of Remark 2.4 and Lemma 3.2, / meets F—D} in a single point on a tip Dt of a rod F—Dv Thence D{=D4 or D6, say D~D4. As in the case (i-A-b), we can show that this contradicts the condition(#), where we take D6 as Dl in the condition(*). Case (ii-A.-b) / f l D i * ^ If D{ is a component of F, we would get a contradiction as in the case (i-Ab). So, we assume that Dx and Dt are contained in distinct connected components of BkD. We may assume i= 7. Let R=D7-\ \-D7+t be a connected component of BkDy which is a rod. If £=0, we would obtain a contradiction to the condition^), where we take D2 as D1 in the condition(*). So, we assume that 2 ^ 1 . Then we have SuppBkD= U ?=iA by virtue of Lemma 3.1, (iii), and R=D7-\-D8. Let ^>=|2/+Z> +z>7|: F->P 1 . We see easily that O is a P^fibration, and that the singular fiber of containing D3\jDi (or D5\JD6, resp.) is given as follows (cf. Lemma 3.3):
1
A
Then we have p(F)^10, which is a contradiction because p(V)=r-\-l—9 (cf. Lemma 3.1). Case (ii-B) / does not meet Bk{D—D^). Then (/, A ) ^ l by Lemma 3.2. It is impossible that (/, A ) ^ 2 (cf. the case (i-B-b)). So, (/, A ) = l - Let a: V-+V be the blowing-up of the point /fi A Let A'=
= U !=iA.
The second csae leads to a contradiction, because (A> 2E-\-D7-\-D^)=l implies (A> A ) = l ° r (A> A ) ^ ! ? while A is an isolated component of BkD. In the first case, we obtain a contradiction to the condition(*), where we take A a s A- Thus, the case (iii) does not occur.
438
D.-Q. ZHANG
Case (iv) F is a fork of type 2?8. From the discussions in the case (ii), we know that it suffices to consider the case: There exists a (—1) curve / such that (/,-Drl \-D^=(l9D^=l9 SuppBfeD=Supp F= U !=iA- a n d D\ is a tip of F as shown below:
-x A
"A.
where
A
A
Consider a Perforation :=|2(/+Di+O2)+C3+D5i: F - > P \ By virtue of Lemma 3.3, the configuration of singular fibers is presented as follows:
If the first case (or the second case, resp.) takes place, we get a contradiction to the condition (*) where we take D5 (or Z)4, resp.) as Dv So, the case (iv) does not occur. Thus, we have verified the claim 1. Q.E.D. Step 2. Our next claim is the following: Claim 2. BkD contains no connected components consisting of three irreducible components. Proof. Suppose that R=D1-\-D?-{-D3 is a connected component of BkD. We assume that D2 is the middle component of R. As in the step 1, there is a (—1) curve / such that either / does not meet Bk(D—D2) or / meets Bk(D—D2) in a single point on a tip Df of a rod Rl9 which is a connected component of Bk(D—D2). We consider these two cases separately. Case (A) / meets Bk(D—D2). Case{K-2) /n2> 2 =^By virtue of Lemma 3.2, R1 is a part of R, whence R1=D1 or D3. This is a contradiction (cf. Lemma 3.2). Case (A-b) IC[D2^F$. If Rx is a part of R, D2 is a tip of R; see the proof for the claim 1, the case
IITAKA SURFACES
439
(i-A-b). This is a contradiction. So Rx is not a part of R, whence R1r\R=. We also have (Z>2, / ) = 1 ; see the case (B) below. Thus, we reach to a contradiction to the condition(*) where we take D{ as Dv Case (B) / does not meet Bk(D-D2). Then (l,D2)^l by virtue of Lemma 3.2. If (l,D2)=l, we reach to a contradiction as in the claim 1, the case (i-A-a). If (/, Z) 2 )^2, one can show, by the arguments in the case (i-B-b) of the claim 1, that D2 is an isolated component of BkD, which is a contradiction. Q.E.D. Step 3. By virtue of the claim 1 and the case(ii), we may assume that BkD contains no forks. We know that r:=#{irreducible components of BkD} = p(V)—1^2 by the hypothesis that V is not isomorphic to P2 or Fm. So, suppose that R is a rod which is a connected component of BkD. Let Dx be a tip of R. As in the proof of the claim 1, there exists a (—1) curve / on V such that either / does not meet Bk{D—D^) or / meets Bk{D—D^) in a single point on a tip Dt of a connected component Rx of Bk(D—D^), which is a rod. We consider these two cases separately. Case (B) / does not meet Bk{D—D^). By virtue of Lemma 3.2, we have {l,D^)^2. We can show that (1,0^= 2, Ar^l-{-Dl9 (A2)=l (whence A is an elliptic curve by Lemma 3.1) and Dx is an isolated component of BkD; see the proof for the claim 1, the case (i-B-b). Let a: V'->V be the blowing-up of the point 1{\A. Then ®\V be the blowing-up of the point /fl A Then 9\a/A\ is an elliptic fibration whose singular fiber is not a multiple fiber, and has one of the configurations listed in the statement of Lemma 3.4.
440
D.-Q.
ZHANG
Now we consider the case where Df is not a component of JR. We may assume that R=D1-\ h A - i a n d R1=Di-\ \-Di+t (t^O). By the assumption that p(V)=l and the condition(*), we see t^3 and t^l. We consider the following cases separately. Namely, Case(a), where R or Rly say i?, consists of more than two components. Hence, by virtue of the claim 2, z'^5. Then, Case(yS), where R and Rx consists of two irreducible components, i.e., i—3 and t=\. We consider first: Case (a) Note that 2=1=2 by virtue of the claim 2. We know that r^6 and r^8(cf. Lemma3.1). We exhibit the configuration of singular fibers of Q\2i+Dl+Di\: V-+P1 as follows: Case i=5. Note that D6 meets E2 and does not meet El by virtue of Lemma 3.2; see the picture below.
X
A AD.
This contradicts the condition(*) where we contract Bk{D—D^). In fact, look at a P^fibration defined by \2{EX+D^+D2+Dt\. Case i ^ 6 , whence r ^ 7 . This case splits to the following three subcases; see the pictures below. In each of these three cases, A is an elliptic curve (cf. Lemma 3.1, (iii)). X-l\-l
(1) r=7 and (A2)=9-r=2. -K
(2) r=8, i=6 and p(F)=r+l=9.
(3) r=8,*=7 and p(F)=r+1=9.
(3)'
441
IITAKA SURFACES
In the second case, we have p(V)^10, which is a contradiction. In the third case, we consider a P^fibration P\ instead of |2/+2?1+i?7|. We present the configuration of singular fibers of \2(E+D3)+D2+D4\ m the picture (3)' above. Then we reach to a contradiction as in the second case above. We now consider the first case. Let a: V'-^V be the blowing-ups of the points If] A and E2 f] A. It is easy to see that V be the blowing-up of the point A'dEi. Let A=
